Fluctuations for Spatially Extended Hawkes Processes

In a previous paper, it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u(t,x) of a neural field equation (NFE). The value u(t,x) represents the membrane potential at time t of a typical neuron located in position x, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u(t,x). Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u(t,x). To the best of our knowledge, this result appears to be new in the literature